Calculus/Limits/Exercises

< Calculus‎ | Limits

Basic Limit ExercisesEdit

1. \lim_{x\to 2}\Big[4x^2-3x+1\Big]

11

2. \lim_{x\to 5}\Big[x^2\Big]

25

Solutions

One-Sided LimitsEdit

Evaluate the following limits or state that the limit does not exist.

3. \lim_{x\to 0^-}\frac{x^3+x^2}{x^3+2x^2}

\frac{1}{2}

4. \lim_{x\to 7^-}\Big[|x^2+x|-x\Big]

49

5. \lim_{x\to -1^+}\sqrt{1-x^2}

0

6. \lim_{x\to -1^-}\sqrt{1-x^2}

The limit does not exist

Solutions

Two-Sided LimitsEdit

Evaluate the following limits or state that the limit does not exist.

7. \lim_{x\to -1}\frac{1}{x-1}

-\frac{1}{2}

8. \lim_{x\to 4}\frac{1}{x-4}

The limit does not exist.

9. \lim_{x\to 2}\frac{1}{x-2}

The limit does not exist.

10. \lim_{x\to -3}\frac{x^2-9}{x+3}

-6

11. \lim_{x\to 3}\frac{x^2-9}{x-3}

6

12. \lim_{x\to -1}\frac{x^2+2x+1}{x+1}

0

13. \lim_{x\to -1}\frac{x^3+1}{x+1}

3

14. \lim_{x\to 4}\frac{x^2+5x-36}{x^2-16}

\frac{13}{8}

15. \lim_{x\to 25}\frac{x-25}{\sqrt{x}-5}

10

16. \lim_{x\to 0}\frac{\left|x\right|}{x}

The limit does not exist.

17. \lim_{x\to 2}\frac{1}{(x-2)^2}

\infty

18. \lim_{x\to 3}\frac{\sqrt{x^2+16}}{x-3}

The limit does not exist.

19. \lim_{x\to -2}\frac{3x^2-8x-3}{2x^2-18}

-\frac{5}{2}

20. \lim_{x\to 2}\frac{x^2 + 2x + 1}{x^2-2x+1}

9

21. \lim_{x\to 3}\frac{x+3}{x^2-9}

The limit does not exist.

22. \lim_{x\to -1}\frac{x+1}{x^2+x}

-1

23. \lim_{x\to 1}\frac{1}{x^2+1}

\frac{1}{2}

24. \lim_{x\to 1}\Big[x^2+5x-\frac{1}{2-x}\Big]

5

25. \lim_{x\to 1}\frac{x^2-1}{x^2+2x-3}

\frac{1}{2}

26. \lim_{x\to 1}\frac{5x}{x^2+2x-3}

The limit does not exist.

Solutions

Limits to InfinityEdit

Evaluate the following limits or state that the limit does not exist.

27. \lim_{x\to\infty}\frac{-x+\pi}{x^2+3x+2}

0

28. \lim_{x\to -\infty}\frac{x^2+2x+1}{3x^2+1}

\frac{1}{3}

29. \lim_{x\to -\infty}\frac{3x^2+x}{2x^2-15}

\frac{3}{2}

30. \lim_{x\to -\infty}\Big[3x^2-2x+1\Big]

\infty

31. \lim_{x\to \infty}\frac{2x^2-32}{x^3-64}

0

32. \lim_{x\to \infty}6

6

33. \lim_{x\to \infty}\frac{3x^2+4x}{x^4+2}

0

34. \lim_{x\to -\infty}\frac{2x+3x^2+1}{2x^2+3}

\frac{3}{2}

35. \lim_{x\to -\infty}\frac{x^3-3x^2+1}{3x^2+x+5}

-\infty

36. \lim_{x\to\infty}\frac{x^2+2}{x^3-2}

0

Solutions

Limits of Piecewise FunctionsEdit

Evaluate the following limits or state that the limit does not exist.

37. Consider the function

f(x)=\begin{cases} (x-2)^2 & \mbox{if }x<2\\x-3 & \mbox{if }x\ge 2. \end{cases}
a. \lim_{x\to 2^-}f(x)

0

b. \lim_{x\to 2^+}f(x)

-1

c. \lim_{x\to 2}f(x)

The limit does not exist


38. Consider the function

g(x)=\begin{cases}-2x+1 & \mbox{if }x\le 0\\x+1 & \mbox{if }0<x<4\\x^2+2 & \mbox{if }x\ge 4. \end{cases}
a. \lim_{x\to 4^+}g(x)

18

b. \lim_{x\to 4^-}g(x)

5

c. \lim_{x\to 0^+}g(x)

1

d. \lim_{x\to 0^-}g(x)

1

e. \lim_{x\to 0}g(x)

1

f. \lim_{x\to 1}g(x)

2


39. Consider the function

 h(x) = \begin{cases} 2x-3 & \mbox{if }x<2\\ 8 & \mbox{if }x=2\\-x+3 & \mbox{if }x>2. \end{cases}
a. \lim_{x\to 0}h(x)

-3

b. \lim_{x\to 2^-}h(x)

1

c. \lim_{x\to 2^+}h(x)

1

d. \lim_{x\to 2}h(x)

1

Solutions

External LinksEdit